Questions C4 - A-Level Maths

OCR MEI C4 2008 January Q8

8 A curve has equation $$x ^ { 2 } + 4 y ^ { 2 } = k ^ { 2 } ,$$ where $k$ is a positive constant.

Verify that $$x = k \cos \theta , \quad y = \frac { 1 } { 2 } k \sin \theta ,$$ are parametric equations for the curve.
Hence or otherwise show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x } { 4 y }$.
Fig. 8 illustrates the curve for a particular value of $k$. Write down this value of $k$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a8332ec-2216-4e1f-9768-ef175c9e159b-4_657_1071_938_577} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
Copy Fig. 8 and on the same axes sketch the curves for $k = 1 , k = 3$ and $k = 4$. On a map, the curves represent the contours of a mountain. A stream flows down the mountain. Its path on the map is always at right angles to the contour it is crossing.
Explain why the path of the stream is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 y } { x } .$$
Solve this differential equation. Given that the path of the stream passes through the point $( 2,1 )$, show that its equation is $y = \frac { x ^ { 4 } } { 16 }$.

OCR MEI C4 2007 June Q1

1 Express $\sin \theta - 3 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R$ and $\alpha$ are constants to be determined, and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Hence solve the equation $\sin \theta - 3 \cos \theta = 1$ for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

OCR MEI C4 2007 June Q2

2 Write down normal vectors to the planes $2 x + 3 y + 4 z = 10$ and $x - 2 y + z = 5$.
Hence show that these planes are perpendicular to each other.

OCR MEI C4 2007 June Q3

3 Fig. 3 shows the curve $y = \ln x$ and part of the line $y = 2$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9296c786-a42a-4aa5-b326-39adbb544cbc-02_250_550_979_753} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} The shaded region is rotated through $360 ^ { \circ }$ about the $y$-axis.

Show that the volume of the solid of revolution formed is given by $\int _ { 0 } ^ { 2 } \pi \mathrm { e } ^ { 2 y } \mathrm {~d} y$.
Evaluate this, leaving your answer in an exact form.

OCR MEI C4 2007 June Q4

4 A curve is defined by parametric equations $$x = \frac { 1 } { t } - 1 , y = \frac { 2 + t } { 1 + t }$$ Show that the cartesian equation of the curve is $y = \frac { 3 + 2 x } { 2 + x }$.

OCR MEI C4 2007 June Q5

5 Verify that the point $( - 1,6,5 )$ lies on both the lines $$\mathbf { r } = \left( \begin{array} { r } 1
2
- 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 0

OCR MEI C4 2007 June Q6

6
3 \end{array} \right) + \mu \left( \begin{array} { r } 1
0
- 2 \end{array} \right)$$ Find the acute angle between the lines. 6 Two students are trying to evaluate the integral $\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x$.
Sarah uses the trapezium rule with 2 strips, and starts by constructing the following table.

$x$	1	1.5	2
$\sqrt { 1 + \mathrm { e } ^ { - x } }$	1.1696	1.1060	1.0655

Complete the calculation, giving your answer to 3 significant figures. Anish uses a binomial approximation for $\sqrt { 1 + \mathrm { e } ^ { - x } }$ and then integrates this.
Show that, provided $\mathrm { e } ^ { - x }$ is suitably small, $\left( 1 + \mathrm { e } ^ { - x } \right) ^ { \frac { 1 } { 2 } } \approx 1 + \frac { 1 } { 2 } \mathrm { e } ^ { - x } - \frac { 1 } { 8 } \mathrm { e } ^ { - 2 x }$.
Use this result to evaluate $\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x$ approximately, giving your answer to 3 significant figures.

OCR MEI C4 2007 June Q7

7 Data suggest that the number of cases of infection from a particular disease tends to oscillate between two values over a period of approximately 6 months.

Suppose that the number of cases, $P$ thousand, after time $t$ months is modelled by the equation $P = \frac { 2 } { 2 - \sin t }$. Thus, when $t = 0 , P = 1$.
1. By considering the greatest and least values of $\sin t$, write down the greatest and least values of $P$ predicted by this model.
2. Verify that $P$ satisfies the differential equation $\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } P ^ { 2 } \cos t$.
An alternative model is proposed, with differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } \left( 2 P ^ { 2 } - P \right) \cos t$$ As before, $P = 1$ when $t = 0$.
1. Express $\frac { 1 } { P ( 2 P - 1 ) }$ in partial fractions.
2. Solve the differential equation (*) to show that $$\ln \left( \frac { 2 P - 1 } { P } \right) = \frac { 1 } { 2 } \sin t$$ This equation can be rearranged to give $P = \frac { 1 } { 2 - \mathrm { e } ^ { \frac { 1 } { 2 } \sin t } }$.
3. Find the greatest and least values of $P$ predicted by this model. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{9296c786-a42a-4aa5-b326-39adbb544cbc-05_609_622_301_719} \captionsetup{labelformat=empty} \caption{Fig. 8}
  \end{figure} In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axes shown, the path of C is modelled by the parametric equations $$x = 10 \cos \theta + 5 \cos 2 \theta , \quad y = 10 \sin \theta + 5 \sin 2 \theta , \quad ( 0 \leqslant \theta < 2 \pi )$$ where $x$ and $y$ are in metres.
4. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { \cos \theta + \cos 2 \theta } { \sin \theta + \sin 2 \theta }$. Verify that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $\theta = \frac { 1 } { 3 } \pi$. Hence find the exact coordinates of the highest point A on the path of C .
5. Express $x ^ { 2 } + y ^ { 2 }$ in terms of $\theta$. Hence show that $$x ^ { 2 } + y ^ { 2 } = 125 + 100 \cos \theta$$
6. Using this result, or otherwise, find the greatest and least distances of C from O . You are given that, at the point B on the path vertically above O , $$2 \cos ^ { 2 } \theta + 2 \cos \theta - 1 = 0$$
7. Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures. \section*{ADVANCED GCE UNIT MATHEMATICS (MEI)} Applications of Advanced Mathematics (C4) \section*{Paper B: Comprehension} \section*{THURSDAY 14 JUNE 2007} Afternoon
  Time: Up to 1 hour
  Additional materials:
  Rough paper
  MEI Examination Formulae and Tables (MF2) Candidate
  Name □
  Centre
  Number sufficient detail of the working to indicate that a correct method is being used. 1 This basic cycloid has parametric equations $$x = a \theta - a \sin \theta , \quad y = a - a \cos \theta$$
  \includegraphics[max width=\textwidth, alt={}]{9296c786-a42a-4aa5-b326-39adbb544cbc-10_307_1138_445_411}
  Find the coordinates of the points M and N , stating the value of $\theta$ at each of them. Point M Point N 2 A sea wave has parametric equations (in suitable units) $$x = 7 \theta - 0.25 \sin \theta , \quad y = 0.25 \cos \theta$$ Find the wavelength and height of the wave.
  3 The graph below shows the profile of a wave.
8. Assuming that it has parametric equations of the form given on line 68 , find the values of $a$ and $b$.
9. Investigate whether the ratio of the trough length to the crest length is consistent with this shape.
  \includegraphics[max width=\textwidth, alt={}, center]{9296c786-a42a-4aa5-b326-39adbb544cbc-11_312_1141_623_415}
10. $\_\_\_\_$
11. $\_\_\_\_$
  4 This diagram illustrates two wave shapes $U$ and $V$. They have the same wavelength and the same height.
  \includegraphics[max width=\textwidth, alt={}, center]{9296c786-a42a-4aa5-b326-39adbb544cbc-12_423_1552_356_205} One of the curves is a sine wave, the other is a curtate cycloid.
12. State which is which, justifying your answer.
13. $\_\_\_\_$
  The parametric equations for the curves are: $$x = a \theta , \quad y = b \cos \theta ,$$ and $$x = a \theta - b \sin \theta , \quad y = b \cos \theta .$$
14. Show that the distance marked $d$ on the diagram is equal to $b$.
15. Hence justify the statement in lines 109 to 111: "In such cases, the curtate cycloid and the sine curve with the same wavelength and height are very similar and so the sine curve is also a good model."
16. $\_\_\_\_$
17. $\_\_\_\_$
  5 The diagram shows a curtate cycloid with scales given. Show that this curve could not be a scale drawing of the shape of a stable sea wave.
  \includegraphics[max width=\textwidth, alt={}, center]{9296c786-a42a-4aa5-b326-39adbb544cbc-13_289_1310_397_331}

OCR MEI C4 2010 June Q1

1 Express $\frac { x } { x ^ { 2 } - 1 } + \frac { 2 } { x + 1 }$ as a single fraction, simplifying your answer.

OCR MEI C4 2010 June Q2

2 Fig. 2 shows the curve $y = \sqrt { 1 + x ^ { 2 } }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c149cb5-7392-4219-b285-486f4694aa6f-2_574_944_612_598} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure}

The following table gives some values of $x$ and $y$.
$x$ 0 0.25 0.5 0.75 1
$y$ 1 1.0308 1.25 1.4142
Find the missing value of $y$, giving your answer correct to 4 decimal places.
Hence show that, using the trapezium rule with four strips, the shaded area is approximately 1.151 square units.
Jenny uses a trapezium rule with 8 strips, and obtains a value of 1.158 square units. Explain why she must have made a mistake.
The shaded area is rotated through $360 ^ { \circ }$ about the $x$-axis. Find the exact volume of the solid of revolution formed.

OCR MEI C4 2010 June Q3

3 The parametric equations of a curve are $$x = \cos 2 \theta , \quad y = \sin \theta \cos \theta \quad \text { for } 0 \leqslant \theta < \pi$$ Show that the cartesian equation of the curve is $x ^ { 2 } + 4 y ^ { 2 } = 1$.
Sketch the curve.

OCR MEI C4 2010 June Q4

4 Find the first three terms in the binomial expansion of $\sqrt { 4 + x }$ in ascending powers of $x$.
State the set of values of $x$ for which the expansion is valid.
show that $\frac { y - 2 } { y + 1 } = A \mathrm { e } ^ { x ^ { 3 } }$, where $A$ is a constant.
(ii) Hence, given that $x$ and $y$ satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } ( y - 2 ) ( y + 1 ) ,$$ 6 Solve the equation $\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }$. Section B (36 marks)
7 A straight pipeline AB passes through a mountain. With respect to axes $\mathrm { O } x y z$, with $\mathrm { O } x$ due East, $\mathrm { O } y$ due North and $\mathrm { O } z$ vertically upwards, A has coordinates $( - 200,100,0 )$ and B has coordinates $( 100,200,100 )$, where units are metres.

OCR MEI C4 2010 June Q6

6 Solve the equation $\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }$. Section B (36 marks)

OCR MEI C4 2010 June Q7

7 A straight pipeline AB passes through a mountain. With respect to axes $\mathrm { O } x y z$, with $\mathrm { O } x$ due East, $\mathrm { O } y$ due North and $\mathrm { O } z$ vertically upwards, A has coordinates $( - 200,100,0 )$ and B has coordinates $( 100,200,100 )$, where units are metres.

Verify that $\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300
100
100 \end{array} \right)$ and find the length of the pipeline.
Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is $x + 2 y + 3 z = 320$.
Find the coordinates of the point where the pipeline meets the layer of rock.
By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer.

OCR MEI C4 2010 June Q8

8 Part of the track of a roller-coaster is modelled by a curve with the parametric equations $$x = 2 \theta - \sin \theta , \quad y = 4 \cos \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ This is shown in Fig. 8. B is a minimum point, and BC is vertical. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c149cb5-7392-4219-b285-486f4694aa6f-4_602_1447_488_351} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

Find the values of the parameter at A and B . Hence show that the ratio of the lengths OA and AC is $( \pi - 1 ) : ( \pi + 1 )$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$. Find the gradient of the track at A .
Show that, when the gradient of the track is $1 , \theta$ satisfies the equation $$\cos \theta - 4 \sin \theta = 2 .$$
Express $\cos \theta - 4 \sin \theta$ in the form $R \cos ( \theta + \alpha )$. Hence solve the equation $\cos \theta - 4 \sin \theta = 2$ for $0 \leqslant \theta \leqslant 2 \pi$. {www.ocr.org.uk}) after the live examination series.
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MATHEMATICS (MEI)} 4754B
Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{Candidates answer on the Question Paper} OCR Supplied Materials:
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\section*{Other Materials Required:}
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Wednesday 9 June 2010 Afternoon
\includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-5_264_456_881_1361} 1 The train journey from Swansea to London is 307 km and that by road is 300 km . Carry out the calculations performed on the First Great Western website to estimate how much lower the carbon dioxide emissions are when travelling by rail rather than road.
2 The equation of the curve in Fig. 3 is $$y = \frac { 1 } { 10 ^ { 4 } } \left( x ^ { 3 } - 100 x ^ { 2 } - 10000 x + 2100100 \right)$$ Calculate the speed at which the car has its lowest carbon dioxide emissions and the value of its emissions at that speed.
[0pt] [An answer obtained from the graph will be given no marks.]
3
In line 109 the carbon dioxide emissions for a particular train journey from Exeter to London are estimated to be 3.7 tonnes. Obtain this figure.
The text then goes on to state that the emissions per extra passenger on this journey are less than $\frac { 1 } { 2 } \mathrm {~kg}$. Justify this figure.
$\_\_\_\_$
$\_\_\_\_$
4 The daily number of trains, $n$, on a line in another country may be modelled by the function defined below, where $P$ is the annual number of passengers. $$\begin{aligned} & n = 10 \text { for } 0 \leqslant P < 10 ^ { 6 }
& n = 11 \text { for } 10 ^ { 6 } \leqslant P < 1.5 \times 10 ^ { 6 }
& n = 12 \text { for } 1.5 \times 10 ^ { 6 } \leqslant P < 2 \times 10 ^ { 6 }
& n = 13 \text { for } 2 \times 10 ^ { 6 } \leqslant P < 2.5 \times 10 ^ { 6 }
& n = 14 \text { for } 2.5 \times 10 ^ { 6 } \leqslant P < 3 \times 10 ^ { 6 }
& \ldots \text { and so on } \ldots \end{aligned}$$
Sketch the graph of $n$ against $P$.
Describe, in words, the relationship between the daily number of trains and the annual number of passengers.
\includegraphics[max width=\textwidth, alt={}, center]{5c149cb5-7392-4219-b285-486f4694aa6f-7_716_1249_1011_440}
$\_\_\_\_$
5 The FGW website gives the conversion factor for miles to kilometres to 7 significant figures.
"We got the distance between the two stations by road from \href{http://theaa.com}{theaa.com}. We then converted this distance to kilometres by multiplying it by $1.609344 . "$ Suppose this conversion factor is applied to a distance of exactly 100 miles.
State which one of the following best expresses the level of accuracy for the distance in metric units, justifying your answer. A : to the nearest millimetre
B : to the nearest 10 centimetres
C : to the nearest metre

OCR MEI C4 2016 June Q1

1 Express $\cos \theta - 3 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$.
Hence show that the equation $\cos \theta - 3 \sin \theta = 4$ has no solution.

OCR MEI C4 2016 June Q2

2 Given that $\left( 1 + \frac { x } { p } \right) ^ { q } = 1 - x + \frac { 3 } { 4 } x ^ { 2 } + \ldots$, find $p$ and $q$, and state the set of values of $x$ for which the expansion is valid.

OCR MEI C4 2016 June Q3

3 Fig. 3 shows the curve $y = x ^ { 4 }$ and the line $y = 4$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b807b2e-777b-4c9a-b3dd-890d21d33174-2_509_510_778_774} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} The finite region enclosed by the curve and the line is rotated through $180 ^ { \circ }$ about the $y$-axis. Find the exact volume of revolution generated.

OCR MEI C4 2016 June Q4

4 Solve the equation $2 \sin 2 \theta = 1 + \cos 2 \theta$ for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.

OCR MEI C4 2016 June Q5

5 In Fig. 5, triangles $\mathrm { ABC } , \mathrm { ACD }$ and ADE are all right-angled, and angles $\mathrm { BAC } , \mathrm { CAD }$ and DAE are all $\theta$. $\mathrm { AB } = x$ and $\mathrm { AE } = 2 x$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b807b2e-777b-4c9a-b3dd-890d21d33174-2_567_465_1905_799} \captionsetup{labelformat=empty} \caption{Fig. 5}

\end{figure}

Show that $\sec ^ { 3 } \theta = 2$.
Hence show the ratio of lengths ED to CB is $2 ^ { \frac { 2 } { 3 } } : 1$.

OCR MEI C4 2016 June Q6

6 P is a general point on the curve with parametric equations $x = 2 t , y = \frac { 2 } { t }$. This is shown in Fig. 6. The tangent at P intersects the $x$ - and $y$-axes at the points Q and R respectively. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b807b2e-777b-4c9a-b3dd-890d21d33174-3_487_684_388_685} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure} Show that the area of the triangle OQR , where O is the origin, is independent of $t$.

OCR MEI C4 Q1

1 Find the coefficient of the term in $x ^ { 3 }$ in the expansion of $\frac { 1 } { ( 2 + 3 x ) ^ { 2 } }$.

OCR MEI C4 Q2

2 The graph shows part of the curve $y = x ^ { 2 } + 1$.
\includegraphics[max width=\textwidth, alt={}, center]{62dbc58e-f498-483f-a9aa-05cb5aa44881-2_380_876_715_575} Find the volume when the area between this curve, the axes and the line $x = 2$ is rotated through $360 ^ { 0 }$ about the $x$-axis.

OCR MEI C4 Q3

3 Solve the equation $\sec ^ { 2 } \theta = 2 \tan \theta + 4$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.

OCR MEI C4 Q4

4 You are given that $\mathbf { a } = \left( \begin{array} { c } 1
2
- 1 \end{array} \right)$ and $\mathbf { b } = \left( \begin{array} { c } 3
- 1
k \end{array} \right)$.

Find the angle between $\mathbf { a }$ and $\mathbf { b }$ when $k = 2$.
Find the value of $k$ such that $\mathbf { a }$ and $\mathbf { b }$ are perpendicular.

Questions C4 (1162 questions)

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OCR MEI C4 2008 January Q8

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OCR MEI C4 Q4

\(x\)	1	1.5	2
\(\sqrt { 1 + \mathrm { e } ^ { - x } }\)	1.1696	1.1060	1.0655

\(x\)	0	0.25	0.5	0.75	1
\(y\)	1	1.0308		1.25	1.4142